Real Spectral Triples over Noncommutative Bieberbach Manifolds

TL;DR

This paper classifies and constructs all real spectral triples over noncommutative Bieberbach manifolds, linking them to classical spin structures and Dirac operators on flat Bieberbach manifolds.

Contribution

It provides a complete classification and explicit construction of real spectral triples over noncommutative Bieberbach manifolds, extending classical geometric concepts to the noncommutative setting.

Findings

Classified all real spectral triples over noncommutative Bieberbach manifolds.

Constructed explicit examples of these spectral triples.

Connected noncommutative geometries to classical spin structures and Dirac operators.

Abstract

We classify and construct all real spectral triples over noncommutative Bieberbach manifolds, which are restrictions of irreducible real equivariant spectral triple over the noncommutative three-torus. We show that in the classical case the constructed geometries correspond exactly to spin structures over Bieberbach manifolds and the Dirac operators constructed for a flat metric.